Question

$overleftrightarrow{pq} perp overleftrightarrow{qr}$ and $overleftrightarrow{tu} perp overleftrightarrow{st}$. complete the proof that $angle stu cong angle pqr$.

	statement	reason
1	$overleftrightarrow{pq} perp overleftrightarrow{qr}$	given
2	$overleftrightarrow{tu} perp overleftrightarrow{st}$	given
3	$mangle pqr = 90^circ$
4	$mangle stu = 90^circ$
5	$mangle pqr = mangle stu$	transitive property of equality
6	$angle stu cong angle pqr$	definition of congruence

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Explanation:

Step1: Analyze Statement 3

To find the reason for \( m\angle PQR = 90^\circ \), we use the definition of perpendicular lines. If two lines are perpendicular, the angle between them is \( 90^\circ \). Since \( \overleftrightarrow{PQ} \perp \overleftrightarrow{QR} \) (from Statement 1), by the definition of perpendicular lines, \( m\angle PQR = 90^\circ \).

Step2: Analyze Statement 4

For \( m\angle STU = 90^\circ \), we use the same logic. Given \( \overleftrightarrow{TU} \perp \overleftrightarrow{ST} \) (from Statement 2), by the definition of perpendicular lines, \( m\angle STU = 90^\circ \).

For Statement 3, the reason is "Definition of perpendicular lines" (because if two lines are perpendicular, the angle between them is \( 90^\circ \), so from \( \overleftrightarrow{PQ} \perp \overleftrightarrow{QR} \), we get \( m\angle PQR = 90^\circ \)).

For Statement 4, the reason is also "Definition of perpendicular lines" (since \( \overleftrightarrow{TU} \perp \overleftrightarrow{ST} \), the angle \( \angle STU \) is \( 90^\circ \) by the definition of perpendicular lines).

Answer:

• For Statement 3: Definition of perpendicular lines
• For Statement 4: Definition of perpendicular lines